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Make use of the Superfish module http://drupal.org/project/superfish for instant drop menus and lots of configuration options. We should create a new region which will be able to host the Superfish Menu block with the appropriate style or any other Menu block.
If there is no Menu block inside the new region, the Corporate Clean will have the current behaviour with CSS drop-down menus. With this way all Drupal installations with Corporate Clean theme could be updated without any problems.

Corporate Clean 7.x-1.6 (and any future release) supports Superfish module. All these changes will be included in the next Corporate Clean release 7.x-1.6 (will be released at the end of this week) and in the next Corporate Clean release 7.x-2.0 with responsive layout (will be released at the end of the next week) - http://drupal.org/node/1655120.

In order to implement the Superfish menus in your Drupal installation with Corporate Clean theme, you should follow this guide.

Install modules

Basic Configuration

6. Superfish module uses the Drupal block system. Therefore, after the installation Superfish module has already created 4 Superfish blocks, which you can access through the Blocks configuration page (Home » Administration » Structure » Blocks). You will notice that the first one has the name "Superfish 1 (Superfish)" and this is by default connected with the Main menu. Click the configure link, set the Title "<none>", enable the "Auto-arrows" option and finally place this block to the "Navigation" region.